Let f : R → R satisfy the equation f ( x + y ) = f ( x ) · f ( y ) for all x , y ∈ R and f ( x ) ≠ 0 for any x ∈ R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h → 0 1 h

Question

Let f : R &#8594; R satisfy the equation f ( x + y ) = f ( x ) &#183; f ( y ) for all x , y &#8712; R and f ( x ) &#8800; 0 for any x &#8712; R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h &#8594; 0 1 h f h - 1 is equal to ___ .

Quantrex Academy · Accepted Answer

Correct answer: 0

If f ( x + y ) = f ( x ) &#183; f ( y ) &#160; &#38; &#160; f ' ( 0 ) = 3 then f ( x ) = a x &#8658; f ' x = a x &#183; l n a &#8658; f ' 0 = l n a = 3 &#8658; a = e 3 &#8658; f x = e 3 x = e 3 x lim x &#8594; 0 f ( x ) - 1 x = lim x &#8594; 0 e 3 x - 1 3 x &#215; 3 = 1 &#215; 3 = 3

Let f : R → R satisfy the equation f ( x + y ) = f ( x ) · f ( y ) for all x , y ∈ R and f ( x ) ≠ 0 for any x ∈ R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h → 0 1 h f h - 1 is equal to ___ .

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let f : R &#8594; R satisfy the equation f ( x + y ) = f ( x ) &#183; f ( y ) for all x , y &#8712; R and f ( x ) &#8800; 0 for any x &#8712; R . If the function f is differentiable at x = 0 and f &#39; ( 0 ) = 3 , then lim h &#8594; 0 1 h f h - 1 is equal to ___ .

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let f : R → R satisfy the equation f ( x + y ) = f ( x ) · f ( y ) for all x , y ∈ R and f ( x ) ≠ 0 for any x ∈ R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h → 0 1 h f h - 1 is equal to ___ .